Financial Management AFIN253
Tutorial 4, Week 5
Homework questions.
A 2 year corporate bond yields 3% pa with a coupon rate of 5% pa, paid semi-annually.
Find the effective monthly rate, effective six month rate, and effective annual rate.
##r_\text{eff monthly}##, ##r_\text{eff 6 month}##, ##r_\text{eff annual}##.
###\begin{aligned} r_\text{eff monthly} &= \left(1+\frac{r_\text{apr comp 6mth}}{2}\right)^{1/6} - 1 \\ &= \left(1+\frac{0.03}{2}\right)^{1/6} - 1 \\ &= 0.002485 \\ \end{aligned}###
###\begin{aligned} r_\text{eff 6mth} &= \frac{r_\text{apr comp 6mth}}{2} \\ &= \frac{0.03}{2} \\ &= 0.015 \\ \end{aligned}###
###\begin{aligned} r_\text{eff annual} &= \left(1+\frac{r_\text{apr comp 6mth}}{2}\right)^{2}-1 \\ &= \left(1+\frac{0.12}{2}\right)^{2}-1 \\ &= 0.030225\\ \end{aligned}###
A 10 year bond has a face value of $100, a yield of 6% pa and a fixed coupon rate of 8% pa, paid semi-annually. What is its price?
###\begin{aligned} P_\text{0} =& \text{PV(annuity of semi-annual coupons)} + \text{PV(face value)} \\ =& C_\text{1,2..T} \times \frac{1}{r_\text{eff 6mth}}\left(1 - \frac{1}{(1+r_\text{eff 6mth})^{T}} \right) + \frac{F_\text{T}}{(1+r_\text{eff 6mth})^{T}} \\ =& \frac{100 \times 0.08}{2} \times \frac{1}{0.06/2}\left(1 - \frac{1}{(1+0.06/2)^{10\times2}} \right) + \frac{100}{(1+0.06/2)^{10 \times 2}} \\ =& 59.50989944 + 55.36757542 \\ =& 114.8774749 \\ \end{aligned} ###
A 60-day Bank Accepted Bill has a face value of $1,000,000. The interest rate is 8% pa and there are 365 days in the year. What is its price now?
###\begin{aligned} P_\text{0, bill} =& \frac{F_d}{1 + r_\text{simple} \times \frac{d}{365}} \\ =& \frac{1,000,000}{1 + 0.08 \times \frac{60}{365}} \\ =& 987020.0108 \\ \end{aligned} ###
You want to buy a house priced at $400,000. You have saved a deposit of $40,000. The bank has agreed to lend you $360,000 as a fully amortising loan with a term of 30 years. The interest rate is 8% pa payable monthly and is not expected to change.
What will be your monthly payments?
Since the interest rate was not specified as an effective annual rate, we can assume that it must be an annualised percentage rate (APR) since by convention and in some countries by law, this is usually the case. Since mortgage loans usually pay interest monthly, by convention the APR can be assumed to compound per month. But to discount the monthly cash flows the effective monthly interest rate is needed, which can be calculated by dividing the annualised percentage rate compounding per month by 12.
###r_\text{eff mthly} = r_\text{APR comp monthly} / 12 = 0.08/12 = 0.00666667###The loan is fully amortising and the interest rate is expected to remain constant so the monthly payments will be equal. We can assume that the payments are made in arrears, as is normal. The annuity equation can be used to discount equal payments:
###\begin{aligned} P_\text{0, fully amortising loan} =& \text{PV(annuity of monthly payments)} \\ =& C_{\text{monthly}} \times \frac{1}{r_\text{eff monthly}} \left(1 - \frac{1}{(1+r_\text{eff monthly})^{T_\text{months}}} \right) \\ 360,000 =& C_{\text{monthly}} \times \frac{1}{0.08/12} \left(1 - \frac{1}{(1+0.08/12)^{30 \times 12}} \right) \\ C_{\text{monthly}} =& 360,000 \div \left(\frac{1}{0.08/12}\left(1 - \frac{1}{(1+0.08/12)^{30 \times 12}} \right) \right) \\ =& 360,000 \div \left(\frac{1}{0.0066667}\left(1 - \frac{1}{(1+0.0066667)^{360}} \right) \right) \\ =& 360,000 \div 136.2834941 \\ =& 2,641.552466 \\ \end{aligned} ###
A share just paid its semi-annual dividend of $5. The dividend is expected to grow at 1% every 6 months forever. This 1% growth rate is an effective 6 month rate.
Therefore the next dividend will be $5.05 in six months. The required return of the stock 8% pa, given as an effective annual rate.
What is the price of the share now?
The dividend is paid every 6 months so we need to discount it using an effective 6 month rate. So first convert the effective annual rate to an effective 6 month rate.
###\begin{aligned} r_\text{eff 6mth} &= (1 + r_\text{eff annual})^{1/2}-1 \\ &= (1 + 0.08)^{1/2}-1 \\ &= 0.039230485 \\ \end{aligned}###
The $5 dividend just paid was received by the previous share owner so we ignore it. The next dividend to be paid will be in 6 months, and it will be 1% bigger than the last. Applying the dividend discount model (DDM),
###\begin{aligned} P_0 &= \frac{C_\text{6mth}}{r_\text{eff 6mth} - g_\text{eff 6mth}} \\ &= \frac{C_\text{0}(1+g_\text{eff 6nth})^1}{r_\text{eff 6mth} - g_\text{eff 6mth}} \\ &= \frac{5(1+0.01)^1}{0.039230485 - 0.01} \\ &= 172.7648405 \\ \end{aligned}###
A share was bought for $4 and paid an dividend of $0.50 one year later (at t=1 year).
Just after the dividend was paid, the share price fell to $3.50 (at t=1 year). What were the total return, capital return and income returns given as effective annual rates? The answer choices are given in the same order:
##r_\text{total}##, ##r_\text{capital}##, ## r_\text{income}##
###\begin{aligned} r_\text{total} =& r_\text{capital} + r_\text{income} \\ =& \frac{P_1 - P_0}{P_0} + \frac{C_1}{P_0} \\ =& \frac{3.50 - 4}{4} + \frac{0.50}{4} \\ =& \frac{-0.5}{4} + \frac{0.5}{4} \\ =& -0.125 + 0.125 \\ \end{aligned}###
So the capital return is -0.125 and the income return is 0.125.
The total return is the sum which is zero:
### r_\text{total} = 0###
A 90-day $1 million Bank Accepted Bill (BAB) was bought for $990,000 and sold 30 days later for $996,000 (at t=30 days).
What was the total return, capital return and income return over the 30 days it was held?
Despite the fact that money market instruments such as bills are normally quoted with simple interest rates, please calculate your answers as compound interest rates, specifically, as effective 30-day rates, which is how the below answer choices are listed.
##r_\text{total}##, ##r_\text{capital}##, ## r_\text{income}##
###\begin{aligned} r_\text{total} =& r_\text{capital} + r_\text{income} \\ =& \frac{P_{30} - P_0}{P_0} + \frac{C_{30}}{P_0} \\ =& \frac{996,000 - 990,000}{990,000} + \frac{0}{990,000} \\ =& \frac{6,000}{990,000} + 0 \\ =& 0.006060606 + 0 \\ \end{aligned}### So the capital return is 0.006060606 and the income return is 0. The total return is the sum which is: ### r_\text{total} = 0.006060606 ###
A company's shares just paid their annual dividend of $2 each.
The stock price is now $40 (just after the dividend payment). The annual dividend is expected to grow by 3% every year forever. The assumptions of the dividend discount model are valid for this company.
What do you expect the effective annual dividend yield to be in 3 years (dividend yield from t=3 to t=4)?
This is a trick question. For stocks whose characteristics can be described by the DDM, the dividend yield is constant through time. This is because a stock's dividend and price are both expected to grow by the same amount, the capital return ##g##, every period. So they are always in the same ratio. Mathematically,
###r_{\text{dividend, }t \rightarrow t+1} = \frac{d_{t+1}}{P_t} = \frac{d_{0}(1+g)^{t+1}}{P_0(1+g)^t} = \frac{d_{1}(1+g)^t}{P_0(1+g)^t} = \frac{d_{1}}{P_0} = r_{\text{dividend, }0 \rightarrow 1}###
###\begin{aligned} r_{\text{dividend, }0 \rightarrow 1} &= \frac{d_{1}}{P_0} \\ &= \frac{d_{0}(1+g)^1}{P_0} \\ &= \frac{2(1+0.03)^1}{40} \\ &= \frac{2.06}{40} = 0.0515 \\ \end{aligned}###
The following equation is the Dividend Discount Model, also known as the 'Gordon Growth Model' or the 'Perpetuity with growth' equation.
###P_0=\frac{d_1}{r-g}###
A stock pays dividends annually. It just paid a dividend, but the next dividend (##d_1##) will be paid in one year.
According to the DDM, what is the correct formula for the expected price of the stock in 2.5 years?
The growth rate in the dividend (##g##) must equal the growth rate in the stock price measured between a whole number of dividend periods. But the growth rate in stock price, also known as the capital return, is actually equal to the total return ##r## in between dividend payments. This is best seen in a saw-tooth graph, where the 'dividend drop-off' price fall can be seen every time a stock pays a dividend. But here it's explained in words:
- The expected capital return measured just after a dividend is paid to just after the next dividend is paid is ##g##.
- The expected capital return measured just before a dividend is paid to just before the next dividend is paid is also ##g##.
- But, the expected capital return measured just after a dividend is paid to just before the next dividend is paid is actually ##r##. The price growth must be higher than ##g## since the stock price must accumulate the next dividend payment as well as the usual price gain over a whole dividend period. Thus the price growth between dividend payments must be ##r = d_1/P_0 + g##.
Using this logic, the growth rate in the share price from just after the current (t=0) dividend was paid to just after the next dividend is paid in one year will be ##g##.
###P_\text{1, just after div} = P_\text{0, just after div}(1+g)^1### Similarly for the next year, just after that dividend is paid (at t=2).###\begin{aligned} P_\text{2, just after div} &= P_\text{1, just after div}(1+g)^1 \\ &= P_\text{0, just after div}(1+g)^2 \\ \end{aligned}###
But from just after the second dividend is paid at t=2 to t=2.5, that period is in between dividend payments, so the share price growth will be the total return ##r##.
###\begin{aligned} P_\text{2.5} &= P_\text{2, just after div}(1+r)^{0.5} \\ &= P_\text{0, just after div}(1+g)^2(1+r)^{0.5} \\ \end{aligned}###
On his 20th birthday, a man makes a resolution. He will deposit $30 into a bank account at the end of every month starting from now, which is the start of the month. So the first payment will be in one month. He will write in his will that when he dies the money in the account should be given to charity.
The bank account pays interest at 6% pa compounding monthly, which is not expected to change.
If the man lives for another 60 years, how much money will be in the bank account if he dies just after making his last (720th) payment?
The effective monthly interest rate can be calculated by dividing the annualised percentage rate compounding per month by 12.
###\begin{aligned} r_\text{eff mthly} &= r_\text{APR comp monthly} / 12 \\ &= 0.06/12 \\ &= 0.005 \\ \end{aligned}###The present value of the annuity of end-of-month payments can be calculated using the ordinary annuity equation. Let the current time at which the man is 20 years old be time zero (t=0).
###\begin{aligned} V_0 &= \frac{C_\text{1, monthly}}{r_\text{eff monthly}}\left(1-\dfrac{1}{(1+r_\text{eff monthly})^{T_\text{months}}}\right) \\ &= \frac{30}{0.005}\left(1-\dfrac{1}{(1+0.005)^{720}}\right) \\ &= 5,834.580469 \\ \end{aligned}###To find the value in 720 months ##(=60\text{ years}\times12\text{ months/year})## we can just future value the present value.
###\begin{aligned} V_\text{T months} &= V_0(1+r_\text{eff monthly})^{T_\text{months}} \\ V_{720} &= 5,834.580469\times(1+0.005)^{720} \\ &= 211,628.4731 \\ \end{aligned}###Bonds X and Y are issued by the same company. Both bonds yield 10% pa, and they have the same face value ($100), maturity, seniority, and payment frequency.
The only difference is that bond X pays coupons of 6% pa and bond Y pays coupons of 8% pa. Which of the following statements is true?
Since bond X's coupon rate (6%) is less than its yield (10%), its price must be less than its face value ($100) so it is a discount bond.
Since bond Y's coupon rate (8%) is less than its yield (10%), its price must be less than its face value ($100) so it is also a discount bond.
A European company just issued two bonds, a
- 3 year zero coupon bond at a yield of 6% pa, and a
- 4 year zero coupon bond at a yield of 6.5% pa.
What is the company's forward rate over the fourth year (from t=3 to t=4)? Give your answer as an effective annual rate, which is how the above bond yields are quoted.
Using the term structure of interest rates equation, also known as the expectations hypothesis theory of interest rates,
###\left(1+r_{\text{0}\rightarrow\text{4yr, eff yrly}}\right)^4 = \left(1+r_{\text{0}\rightarrow\text{3yr, eff yrly}}\right)^3 \left(1+r_{\text{3}\rightarrow\text{4yr, eff yrly}}\right)^1 ### ###\left(1+0.065\right)^4 = \left(1+0.06 \right)^3 \left(1+r_{\text{3}\rightarrow\text{4yr, eff yrly}}\right)^1 ### ###1+r_{\text{3}\rightarrow\text{4yr, eff yrly}} = \frac{\left(1+0.065\right)^4}{\left(1+0.06 \right)^3} ### ###\begin{aligned} r_{\text{3}\rightarrow\text{4yr, eff yrly}} &= \frac{\left(1+0.065\right)^4}{\left(1+0.06 \right)^3} - 1\\ &= 0.080141955 \\ \end{aligned} ###You're trying to save enough money for a deposit to buy a house. You want to buy a house worth $400,000 and the bank requires a 20% deposit ($80,000) before it will give you a loan for the other $320,000 that you need.
You currently have no savings, but you just started working and can save $2,000 per month, with the first payment in one month from now. Bank interest rates on savings accounts are 4.8% pa with interest paid monthly and interest rates are not expected to change.
How long will it take to save the $80,000 deposit? Round your answer up to the nearest month.
Since the interest rate was not specified as an effective annual rate, we can assume that it must be an annualised percentage rate (APR) since by convention and in some countries by law, this is usually the case. Since the bank account pays interest monthly, by convention the APR can be assumed to compound per month. But to discount the monthly cash flows the effective monthly interest rate is needed, which can be calculated by dividing the annualised percentage rate compounding per month by 12.
###\begin{aligned} r_\text{eff mthly} &= r_\text{APR comp monthly} / 12 \\ &= 0.048/12 \\ &= 0.004 \\ \end{aligned}###The future value of the annuity of end-of-month payments can be calculated using the ordinary T-period annuity equation, but grown ahead by the number periods T.
###V_0 = \frac{C_\text{1, monthly}}{r_\text{eff monthly}}\left(1-\dfrac{1}{(1+r_\text{eff monthly})^{T_\text{months}}}\right)### ###V_T = \frac{C_\text{1, monthly}}{r_\text{eff monthly}}\left(1-\dfrac{1}{(1+r_\text{eff monthly})^{T_\text{months}}}\right) (1+r_\text{eff monthly})^{T_\text{months}}### ###80,000 = \frac{2,000}{0.004}\left(1-\dfrac{1}{(1+0.004)^{T}}\right) (1+0.004)^{T}### ###80,000 = \frac{2,000}{0.004}\left((1+0.004)^{T}-1\right) ### ###(1+0.004)^{T} = \frac{80,000 \times 0.004}{2,000}+1 ### ###\ln{\left((1+0.004)^{T}\right)} = \ln{\left( \frac{80,000 \times 0.004}{2,000}+1 \right)} ### ###T.\ln{(1+0.004)} = \ln{\left( \frac{80,000 \times 0.004}{2,000}+1 \right)} ### ###\begin{aligned} T &= \frac{ \ln{\left( \frac{80,000 \times 0.004}{2,000}+1 \right)} }{ \ln{(1+0.004)} } \\ &= 37.17916191 \\ \end{aligned}###Since you only have the cash at the end of each month, round the decimal number of months up to get 38 months.
A student won $1m in a lottery. Currently the money is in a bank account which pays interest at 6% pa, given as an APR compounding per month.
She plans to spend $20,000 at the beginning of every month from now on (so the first withdrawal will be at t=0). After each withdrawal, she will check how much money is left in the account. When there is less than $500,000 left, she will donate that remaining amount to charity.
In how many months will she make her last withdrawal and donate the remainder to charity?
The effective monthly interest rate can be calculated by dividing the annualised percentage rate compounding per month by 12.
###\begin{aligned} r_\text{eff mthly} &= r_\text{APR comp monthly} / 12 \\ &= 0.06/12 \\ &= 0.005 \\ \end{aligned}###There are three main cash flow terms that need to be included in our analysis:
- The $1 million in the bank account at the start (##V_0##).
- The $0.02 million withdrawn at the start of every month for T months (##C_0 = C_1=C_2=...##).
- The $0.5 million remaining in the bank account in T months (##V_T##). We actually want to find the first time that the account has less than $0.5m. But this will be the time of the next withdrawal after the time that we calculate there is exactly $0.5m.
Now to set up the pricing equation. We can take the present value of each cash flow or the future value or anything in between. It doesn't matter. Here we will find the future value of all cash flows in T months.
- The future value of the $1 million (##V_0##) is easy, just grow it forward by T months.
- The future value of the monthly $0.02 million payments (##C_0 = C_1=C_2=...##) can be calculated using the ordinary T-period annuity equation grown ahead by one period to adjust for how the payments are at the start of each month. This whole amount is grown again by T periods to get the future value.
- The $0.5 million left in the bank account in T months (##V_T##) is already a future value at T months so there is no discounting needed.
A decimal answer is not feasible since the withdrawals only happen at the start of each month, so round the answer up to get 31 months which is when the balance in the account will be less than $500,000 for the first time. There will actually be $495,034.6084 in the account at that time.
You just started work at your new job which pays $48,000 per year.
The human resources department have given you the option of being paid at the end of every week or every month.
Assume that there are 4 weeks per month, 12 months per year and 48 weeks per year.
Bank interest rates are 12% pa given as an APR compounding per month.
What is the dollar gain over one year, as a net present value, of being paid every week rather than every month?
The difference in present values between getting paid at the end of every week and the end of every month can be calculated by subtracting a monthly annuity by a weekly annuity. But first we need to convert the yield to an effective weekly rate and an effective weekly rate and an effective monthly rate:
The difference in present values between getting paid at the end of every week and the end of every month can be calculated by subtracting a monthly annuity by a weekly annuity. But first we need to convert the yield to an effective weekly rate and an effective weekly rate and an effective monthly rate:###\begin{aligned} r_\text{eff monthly} &= \frac{r_\text{apr comp monthly}}{12} \\ &= \frac{0.12}{12} \\ &= 0.01 \\ \end{aligned}###
Since we can assume that there are 4 weeks per month,
###\begin{aligned} r_\text{eff weekly} &= \left(1 + \frac{r_\text{apr comp monthly}}{12} \right)^{1/4} - 1 \\ &= \left(1 + \frac{0.12}{12} \right)^{1/4} - 1 \\ &= 0.002490679 \\ \end{aligned}###
To calculate the present value difference between the weekly and monthly annuities,
###\begin{aligned} V_\text{0, gain} =& \frac{C_\text{1,weekly}}{r_\text{eff weekly}}\left(1 - \frac{1}{(1+r_\text{eff weekly})^\text{weeks in yr}} \right) - \frac{C_\text{1,monthly}}{r_\text{eff monthly}}\left(1 - \frac{1}{(1+r_\text{eff monthly})^\text{months in yr}} \right) \\ =& \frac{48,000/48}{0.002490679 }\left(1 - \frac{1}{(1+0.002490679 )^{48}} \right) - \frac{48,000/12}{0.01}\left(1 - \frac{1}{(1+0.01)^{12}} \right) \\ =& 45,188.78608 - 45,020.30989 \\ =& 168.4761885 \\ \end{aligned}###
Your main expense is fuel for your car which costs $100 per month. You just refueled, so you won't need any more fuel for another month (first payment at t=1 month).
You have $2,500 in a bank account which pays interest at a rate of 6% pa, payable monthly. Interest rates are not expected to change.
Assuming that you have no income, in how many months time will you not have enough money to fully refuel your car?
The payments are monthly so let's first convert the interest rate, which must be an APR compounding per month, to an effective monthly rate.
###r_\text{eff monthly} = r_\text{apr comp monthly}/12 = 0.06/12 = 0.005###
The annuity equation can be used to find the time when the $2,500 runs out after repeated $100 withdrawals at the end of the month.
###V_0 = C_\text{1, monthly} \times \frac{1}{r_\text{eff mthly}} \left( 1 - \frac{1}{(1+r_\text{eff mthly})^{T}} \right) ### ###2,500 = 100 \times \frac{1}{0.005} \left( 1 - \frac{1}{(1+0.005)^{T}} \right) ### ###\frac{2,500 \times 0.005}{100} = 1 - \frac{1}{(1+0.005)^{T}} ### ###(1+0.005)^{-T} = 1 - \frac{2,500 \times 0.005}{100} ### ###\ln\left((1+0.005)^{-T}\right) = \ln\left(1 - \frac{2,500 \times 0.005}{100}\right) ### ###-T . \ln\left(1+0.005\right) = \ln\left(1 - \frac{2,500 \times 0.005}{100}\right) ######\begin{aligned} T &= -\frac{\ln\left(1 - \frac{2,500 \times 0.005}{100}\right)}{\ln\left(1+0.005\right)} \\ &= 26.77298872 \text{ months} \\ &= 27 \text{ months, rounded up to get the feasible solution.} \\ \end{aligned}###
The decimal answer is not a feasible solution since we only refuel at the end of each month, not at a fractional time period 0.77298872 way through the month. Therefore we round this number up to get the first time that we go to the petrol station and can not afford to fully refuel. This will be at t=27, the end of the 27th month.
In these tough economic times, central banks around the world have cut interest rates so low that they are practically zero. In some countries, government bond yields are also very close to zero.
A three year government bond with a face value of $100 and a coupon rate of 2% pa paid semi-annually was just issued at a yield of 0%. What is the price of the bond?
When the yield is zero, there is no time value of money. Therefore we can just sum cash flows like an accountant. Over the 3 year bond's maturity there will be 6 semi-annual coupon payments of $1 each, and the face value paid at maturity.
###\begin{aligned} P_\text{0, bond} &= 6 \times C + F \\ &= 6 \times 1 + 100 = 106 \\ \end{aligned}###
Interestingly, the normal way to value a fixed-coupon bond using the annuity equation will not work since there will be a divide by zero problem which is mathematically impossible:
###\begin{aligned} P_0 &= C_\text{1} \times \frac{1}{r_\text{eff 6mth}} \left( 1 - \frac{1}{(1+r_\text{eff 6mth})^{T}} \right) + \frac{F_T}{(1+r_\text{eff 6mth})^T} \\ &= 1 \times \color{red}{\frac{1}{0}} \left( 1 - \frac{1}{(1+0)^{6}} \right) + \frac{100}{(1+0)^6} \\ \end{aligned}###
Which is mathematically undefined, so that is a dead-end.
But present-valuing the individual payments separately will still work.
###\begin{aligned} P_0 &= \frac{C_\text{0.5 yr}}{(1+r_\text{eff 6mth})^1} + \frac{C_\text{1 yr}}{(1+r_\text{eff 6mth})^2} + \frac{C_\text{1.5 yr}}{(1+r_\text{eff 6mth})^3} + \frac{C_\text{2 yr}}{(1+r_\text{eff 6mth})^4} +\frac{C_\text{2.5 yr}}{(1+r_\text{eff 6mth})^5} + \frac{C_\text{3 yr}}{(1+r_\text{eff 6mth})^6} + \frac{F_\text{3 yr}}{(1+r_\text{eff 6mth})^6} \\ &= \frac{1}{(1+0)^1} + \frac{1}{(1+0)^2} + \frac{1}{(1+0)^3} + \frac{1}{(1+0)^4} +\frac{1}{(1+0)^5} + \frac{1}{(1+0)^6} + \frac{100}{(1+0)^6} \\ &= 1+1+1+1+1+1+100 \\ &= 6 \times 1 + 100 \\ &= 106 \\ \end{aligned}###
You own an apartment which you rent out as an investment property.
What is the price of the apartment using discounted cash flow (DCF, same as NPV) valuation?
Assume that:
- You just signed a contract to rent the apartment out to a tenant for the next 12 months at $2,000 per month, payable in advance (at the start of the month, t=0). The tenant is just about to pay you the first $2,000 payment.
- The contract states that monthly rental payments are fixed for 12 months. After the contract ends, you plan to sign another contract but with rental payment increases of 3%. You intend to do this every year.
So rental payments will increase at the start of the 13th month (t=12) to be $2,060 (=2,000(1+0.03)), and then they will be constant for the next 12 months.
Rental payments will increase again at the start of the 25th month (t=24) to be $2,121.80 (=2,000(1+0.03)2), and then they will be constant for the next 12 months until the next year, and so on. - The required return of the apartment is 8.732% pa, given as an effective annual rate.
- Ignore all taxes, maintenance, real estate agent, council and strata fees, periods of vacancy and other costs. Assume that the apartment will last forever and so will the rental payments.
This question can be calculated in three steps.
First find the total required return as an effective monthly rate since the rental payments are monthly.
###r_\text{eff mthly} = (1+r_\text{eff yrly})^{1/12} - 1 = (1+0.08732)^{1/12} - 1 = 0.007000721###
Second, find the present value of a year's worth of rental payments which can be done with an annuity of the 12 equal monthly payments. Note that the annuity's first payment is just about to occur right now (at t=0). The annuity equation gives a value one period before (at t=-1 month), so we grow it forward by one period to get the present value (at t=0), which makes what some call the 'annuity due' formula.
###\begin{aligned} V_\text{0, 12 months rent} &= \dfrac{C_\text{0, mthly}}{r_\text{mthly}} \left(1- \dfrac{1}{(1+r_\text{mthly})^{12}} \right) (1+r_\text{mthly})^1 \\ &= \dfrac{2,000}{0.007000721} \left(1- \dfrac{1}{(1+0.007000721)^{12}} \right) (1+0.007000721)^1 \\ &= 23,103.2659 \\ \end{aligned}###
Third, this present value of annual rental payments will grow by 3% pa forever, so we can now discount this using the perpetuity formula. Again we have to take care since the annual rental value is a present value (at t=0) and the perpetuity gives a value one period before (at t=-1 year) so we grow it forward by one year to get the present value of the rental payments in perpetuity.
###\begin{aligned} P_\text{0} &= \left( \frac{ V_\text{0, 12 months rent} }{r_\text{yrly}-g_\text{yrly}} \right) (1+r_\text{yrly})^1\\ &= \left( \frac{ 23,103.2659 }{0.08732-0.03} \right) (1+0.08732)^1\\ &= 438,252.6707 \\ \end{aligned}###
Combining all steps in one big formula,
###\begin{aligned} P_\text{0} &= \left( \vcenter{ \frac{ \dfrac{C_\text{0, mthly}}{r_\text{mthly}} \left(1- \dfrac{1}{(1+r_\text{mthly})^{12}} \right) (1+r_\text{mthly})^1 }{r_\text{yrly}-g_\text{yrly}} } \right) (1+r_\text{yrly})^1\\ &= \left( \vcenter{ \frac{ \dfrac{2,000}{0.007000721} \left(1- \dfrac{1}{(1+0.007000721)^{12}} \right) (1+0.007000721)^1 }{0.08732-0.03} } \right) (1+0.08732)^1\\ &= \left( \frac{ 23,103.2659 }{0.08732-0.03} \right) (1+0.08732)^1\\ &= 438,252.6707 \\ \end{aligned}###
This formulaic model can be made a little more elegant by not requiring the monthly effective total return as follows:
###\begin{aligned} P_\text{0} &= \vcenter{ \frac{ \left( \dfrac{C_\text{0, mthly}}{(1+r_\text{yrly})^{1/12}-1} \right) \left(1- \dfrac{1}{(1+r_\text{yrly})^{1}} \right) (1+r_\text{yrly})^{13/12} }{r_\text{yrly}-g_\text{yrly}} } \\ \end{aligned}###
Another way to approach this question is to make an annuity of perpetuities. Think of the monthly payments as 12 separate perpetuities, one for each month, that are received each year and increase by the annual growth rate. These 12 perpetuities will all be equal so they can be present valued using the annuity formula.
###\begin{aligned} P_\text{0} &= \left(\dfrac{C_\text{0, yrly}}{r_\text{yrly}-g_\text{yrly}}\right)(1+r_\text{yrly})^1 { \dfrac{ 1 }{r_\text{mthly}} \left(1- \dfrac{1}{(1+r_\text{mthly})^{12}} \right) (1+r_\text{mthly})^1 } \\ &= \left(\dfrac{2,000}{0.08732-0.03}\right)(1+0.08732)^1 { \dfrac{ 1 }{0.007000721} \left(1- \dfrac{1}{(1+0.007000721)^{12}} \right) (1+0.007000721)^1 } \\ &= 34,891.83531 \times (1+0.08732)^1 \times 11.47132541 \times (1+0.007000721)^1 \\ &= 438,252.6707 \\ \end{aligned}###
Comments
This question is interesting because it values property using discounted cash flows, whereas most property valuers and real estate agents use a comparable sales or multiples approach to valuing property.
- Comparable sales: find properties that have sold recently with similar characteristics and take an average of their sale prices.
- Multiples valuation: a rough and simple measure of how much a property is worth. An example that is often used in Sydney Australia is to multiply the weekly rent of the property by 1,000 to get the price. So a property that rents for $2,000/month equates to about $500 per week, so this property would be worth around $500,000. Of course this is just a rule of thumb used to make quick estimates.
The discounted cash flow valuation used here depends heavily on the inputs. A small decrease in annual forecast rental growth, or a small increase in the total required return will lower the price considerably. These variables have a non-linear effect on price.
The ongoing costs of renting such as agent fees and maintenance have a linear effect on the value. If half of the gross monthly rent was spent on ongoing costs, then the property price would be half as much.